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The Quintessential Elements
Ian Beardsley (Oct 6, 2021)!
Physics, University of Oregon
Genesis Project California 2021
⋅
⋅
of 2 32
1. Abstract We demonstrate the periodic table for artificial intelligence (AI) elements Arsenic
to Iron either elements of semiconductors for transistor technology or conductors for
electrical wire, adhere to a musical scheme. We want to describe life in terms of these AI
elements and show they are both mathematical constructs. The relationship between notes
by music theory, is a mathematical construct. Elements are shown to be in five-fold (fifth
root) relation making them quintessential.!
2. Nickel and Beryllium Alpha decay is the decay of elements where a helium nucleus is
ejected from an element (alpha particles). This happens with atoms heavier than nickel
because the atom has to be heavy enough that the internal repulsion of protons is high enough
that the binding force has trouble holding them together. However, interestingly beryllium-8 is
an exception; it is much lighter than nickel. Beryllium-8 was determined by Fred Hoyle to make
carbon in nuclear synthesis in stars by combining with helium. He sought to find the process
by which carbon is created because the Universe needs to make it if we are to have life as we
know it. !
3. The Theory We pull these AI elements out of the periodic table of the elements to make an
AI periodic table:!
We now notice we can make a 3 by 3 matrix of it, which lends itself to to the curl of a vector
field, by including biological elements carbon C (above Si):!
=!
=!
!
Which resulted in Stokes theorem (Beardsley, Essays In Cosmic Archaeology Volume 3):!
i
j
k
∂
∂x
∂
∂y
∂
∂z
(−C ⋅ P)y (Si ⋅ Ga)z (Ge ⋅ As)y
(Ge ⋅ As − Si ⋅ G a)
i + (C ⋅ P)
k
[
(72.64)(74.92) − (28.09)(69.72)
]
i +
[
(12.01)(30.97)
]
k
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5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = ex p
(
1
Ge − Si
∫
Ge
Si
ln(x)d x
)
∇ ×
u = (Ge ⋅ As − Si ⋅ Ga)
i + (C ⋅ P)
k
d
a =
(
zdydz
i + yd ydz
k
)
u = (−C ⋅ P)y
i + (Si ⋅ Ge)z
j + (Ga ⋅ As)y
k
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
n
n
∏
i=1
x
i
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!
=(28.085)(72.64)(12.085)(107.8682)(196.9657)=!
!
Where we have substituted carbon (C=12.01) the core biological element for copper (Cu).!
But since we have:!
!
1
We take the ratio and have!
!
Almost exactly 3 which is the ratio of the perimeter of regular hexagon to its diameter used to
estimate pi in ancient times by inscribing it in a circle:!
!
Perimeter=6!
Diameter=2!
6/2=3!
5
∏
i=1
x
i
= Si ⋅ Ge ⋅ Cu ⋅ Ag ⋅ Au
523,818,646.5
g
5
mol
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = 170,535,359.662(g/m ol )
5
523,818,646.5
170,535,359.662
= 3.0716
See Appendix 1
1
of 5 32
!
Thus we have the following equation…!
!
4. Musical Intervals The western tonal system divides up the tonic to double its frequency in
12 tones. But the musical key is defined by the pattern that is only eight of these tones. Thus
at times we must skip a tone (a note). In fact, to comprise the tones of a major or minor key we
play the maximum number of whole steps (the skipping of a note). Since twelve divided by two
is six and we need eight tones two of them must be a half step (where we don’t skip a note). In
the major key this is wholestep—wholestep—halfstep. The minor key begins wholestep-
halfstep.!
We guess that the periodic table of the elements is music (See Figure 1). To do this I am
interested in starting at arsenic (As) because it is a semiconductor doping agent. We then skip
germanium (Ge) an element doped with arsenic to make negative (n-type) germanium, and go
to the next element to the left after the germanium to have gone a whole step and landed on
the other doping agent gallium (Ga) which in contrast makes positive (p-type) germanium. Then
we continue with another whole step to make a major key and it is copper (Cu) the most
abundant practical conductor used for making electrical wire. Now we have to go a half a step
which takes us to nickel (Ni) which like copper is perfect because it is the element beyond
which alpha decay can happen (emits) an alpha particle (helium nucleus) when heated
(Beryllium an exception). Now we go a whole step as is done in the major key and this lands us
on iron (Fe). This is good because stars on the main sequence have luminosities in terms solar
luminosities L related to their mass M in solar masses given by!
!
But we notice the molar mass of gold (Au) divided by iron (Fe) is 3.5 yielding!
"
π = 3.141...
π
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
5
∏
i=1
x
i
L = M
3.5
L = M
Au
Fe
Figure 1: The elements as musical intervals.
of 6 32
We see whether or not we take a whole step in the periodic table for these elements we get a
ratio between successive terms that is always around 107.635. We look at the musical scale
taking C major since the notes are all natural. We have!
C. D. E. F. G. A. B. C.!
We have the halfstep - between E and F - is 106% where for the whole steps it is 112.25%.
The geometric mean between these gives!
!
Which is very close to our 107.635 and we have!
109.08024-107.635=1.44524!
(109.08024)/107.635=1.013427231!
Why look at these elements by molar mass in terms of musical notes? By making the halfsteps
that change in pitch by the same amount for each step as well as these half steps being such
that 12 of them is an octave, then they land on frequencies such that every other note in a
scale makes the interval of a third and three of them defining what is called a triad and they
land on harmonic intervals such as the 4th, and the 5th (See Figure 2). These add more
constructively as vibrating waves than say other intervals that are more dissonant. We can
relate these changes in frequency to changes in string length such as 3/4 and 2/3 which may
have something to offer in terms of physical construction of matter: We see our half step for the
elements occurs for nickel to copper, which is what we want!"
(106)(112.25) = 109.0802457
Figure 2: String length as related to frequency.
of 7 32
5. Elemental Scales So let’s do a more thorough analysis with increase in note frequency
increasing as molar mass frequency increases (See Figures 3 and 4). We have!
!
!
!
We see here the result the ratios between successive elements by molar mass is about the
same as successive half steps between notes (cycles per second). In a minor scale we start at
Fe then skip cobalt (Co) to make a whole step to nickel (Ni). This is a percent change of!
!
We see the whole step from Ni to Fe is 105% about a musical half step like C/B=106%. For
this to be a minor scale the next step is a half step from Ni to Cu, which gives!
!
Our whole steps are E/D=112% and (112+106)/2=109% putting the elemental half step of !
Cu/Ni the average between a musical whole step and a musical half step. In the minor
pentatonic we have whole steps and an interval of a minor third. This is!
!
Thus the minor third for these elements is about a musical whole step (112).!
We see choosing Iron (Fe) is wise as our starting point because as we said for a main sequence
star!
!
And, this makes the last element krypton which being in the last column of the periodic table
(group 18) is an inert gas — so it is like the the note C which is the reference point for music in
the western system of music theory. Krypton being in group 18 and an inert gas because of
this, is an elemental reference point in the periodic table because for example for carbon (C) we
have 18-14=4 valence electrons to acquire noble gas electron configuration. Our pentatonic
scale is the notes!
Kr
Br
=
83.80
79.90
= 104.88
C
B
=
523.25
493.88
= 105.9467887
Br
Si
=
79.90
78.96
= 101.19
B
A
♯
=
49.88
466.164
= 105.9455471
Se
As
=
78.96
74.92
= 105.39
A
♯
A
=
466.164
440
= 105.946
Ni
Fe
=
58.69
55.85
= 105.085
E
D
=
329.628
293.665
= 112.25
Cu
Ni
=
63.55
58.69
= 108.28
F
E
=
349.228
329.628
= 106
Zn
Co
=
F
♯
D
♯
⟶
65.39
58.93
= 111 =
369.944
311.127
= 119
L = M
Au
Fe
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D sharp, F sharp, G sharp, A sharp,…!
Which are elements!
Co, Zn, Ge, Se,…!
And these are the notes outside of the minor scale beginning at Fe. But we can start our minor
pentatonic on Fe, which corresponds to a different key. We have minor pentatonic 2 is:!
D, E, F, G, A,…!
Which are elements!
Fe, Cu, Ga, As,…!
We find that in both pentatonic scales, 1 and 2 the intervals of a minor third (Zn/Co and Cu/Fe)
are very close to the musical whole steps which are are a changes of 112%. Figures 3 and 4…"
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"
Figure 3: Molar mass and Hz
of 10 32
!
Figure 4: Minor pentatonics in two different keys.
of 11 32
6. Length and Frequency
Reduce the string length to 2/3 its open length and that is an interval of a fifth. That is a change
in frequency of D=293.665 Hz to B=493.88 Hz. Reduce the string length to 3/4 its open length
and that is a change in frequency from D=293.665 Hz to A=440 Hz, which is an interval of a
fourth. Cut the string length in half and you double the frequency. Which is an octave. We have!
!
!
!
Where is the golden ratio and is the golden ratio conjugate. And,…!
!
Thus, we see string length is inversely proportional to frequency. There are two equations for
string length, which we can examine by looking at a guitar (See figure 5). From the bridge of a
guitar to the fret, where the open string length is s we have:!
!
And!
!
Where s is the string length from the bridge to the nut and the nut is , fret 1 is , fret 2 is ,…!
is the distance from , and is the distance from , and so on…!
The 3/2 is an approximation to the the golden ratio (Phi).!
We can think of electron orbits in a hydrogen atom as frequencies related to length as well
because for a drop in orbit a photon is emitted that has a frequency associated with it. The
change in orbit is like the change in the length of a string. Since the n=3 orbit is a distance of
R3=4.761E-10m from the nucleus and R2=2.116E-10m we have R3/R2=2.25. But the Energy at
E3 is E3=-13.6eV/9, and at E2 it is -13.6eV/4. This is a ratio of 4/9=0.444 and 1/0.444==2.25,
thus there is an inverse relationship between length and frequency in the atom (See Fig. 5).!
1
2
= 2
2
3
=
493.88
391.995
= 1.681 = Φ
Φ =
5 + 1
2
=
1
ϕ
, ϕ =
5 − 1
2
≈ 2/3
Φ
ϕ
3
4
=
440
293.665
= 1.2599 =
4
3
= 1.5
l =
s
2
n/12
l
i
=
s
i
17.817
l
0
l
1
l
2
l
1
l
0
l
2
l
1
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"
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7. Discussion
I find this equation very beautiful!
!
Because on the left side we have what I call the AI matrix:!
!
Because it builds the u vector!
!
And and on the right side we have the what I call the electronics matrix:!
And in that on the left we have a double integral multiplied by pi, over an area and on the right
we have a product operator. Essentially the beauty is in the integral calculus on the left
connecting the AI matrix to the electronics matrix on the right with the product calculus. We
have to ask why this holds for the molar masses of these elements. It has a geometric
representation in that it is flux of the curl of a vector field through a surface with the product of
five elements on the right. This is built from Stokes theorem for these semiconductor and
electronics matrices. What could have brought about the physical properties of these elements
such that this relationship between surface and line holds which we see in its manifestation as!
!
!
π
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
5
∏
i=1
x
i
u = (−C ⋅ P)y
i + (Si ⋅ Ge)z
j + (Ga ⋅ As)y
k
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = ex p
(
1
Ge − Si
∫
Ge
Si
ln(x)d x
)
5
∏
i=1
x
i
= Si ⋅ Ge ⋅ C ⋅ Ag ⋅ Au
of 14 32
Where we have the finest transistor elements Si and Ge to the the left of the core biological life
element C and on the right of it the finest conductors for electronics wire Ag, and Au. What
force was behind the creation of these elements such that these theorems hold for them, and
does it serve a purpose, and if it does is there a reason for it?!
This question has been the reason for finding in this paper how the atoms which are elements
might be music theory, because these equations seem to be music.!
8. Conclusion: We may be able to explain the structure of matter at its basis - the elements -
in terms of music theory. The reason perhaps the electron orbits might be described by music
is that musical intervals are in ratios that don’t interfere with one another, which may what is
meant by sonority. Further In our finding of a Stokes theorem representation of semiconductor
and electronics matrices we see the categories of biological and electronics elements may be
mathematical constructs not just chemical. Why?!
They seem to be reconciling!
!
For a hexagon (a is its apothem, P its perimeter) with!
!
For a circle, in that!
!
!
As we show using the most accurate data available in Appendix 1 which are accurate to at
least 3 places after the decimal except germanium."
A =
1
2
aP
A = π r
2
523,818,646.5
170,535,359.662
= 3.0716
3.141 + 3.00
2
= 3.0705
of 15 32
!
Plot of the surface area
of 16 32
9. Atomic Number A very convenient way to estimate pi is with the regular hexagon because
its side is the same as its radius. Thus if its side is one then its radius is one meaning its
permitter is six and its diameter is two giving the integer three even:!
!
We see this can very accurately be approximated by averaging the regular
pentagon with the regular octagon.!
!
The sum of the angles is . .
, , and 54+36=90 where
. a=0.68819096. We have and
then the diameter D is .!
The perimeter P over the diameter is . By similar reasoning we
have for a regular octagon:!
!
. The angle !
Thus,…!
!
Is a regular hexagon.!
We see that the atomic radio of silicon the core element of artificial intelligence (transistor
technology) fits together with the core element of biological life carbon if the silicon is taken as
inscribed in a regular dodecagon and the carbon is taken as inscribed in a regular octagon. We
have:!
, , !
, , !
Apothem: "
A = 180
∘
(n − 2)
A = 180
∘
(3) = 540
∘
540
∘
5
= 108
∘
108
∘
2
= 54
∘
a /s
2
= tan54
∘
2 cos36
∘
= Φ
a
2
+ (s /2)
2
= 0.850650808
D = 1.701301617 ≈ 3
P/D = 2.938926261 ≈ 3
P/D = 3.061457459 ≈ π
22.5
∘
= 0.41421 ≈ 2 − 1
3.061407459 + 2.93892621
2
= 3.00019686 ≈ = 3.00000 = 3
D = 1 + 2x
x
2
+ x
2
= 1
2
2x
2
= 1
2x
2
= 1
x = 2/2
D = 1 + 2
a = (1 + 2)/2 = 1.2071
of 17 32
For a regular dodecagon:!
!
The radius of a silicon atom is Si=0.118nm and that of carbon is
C=0.077nm:!
!
!
!
This has an accuracy %!
It makes sense that we define molar mass in terms of carbon because being element six it can
be thought of in terms of our regular hexagon which defines pi as an integer, the integer 3 (that
is to say as a whole number, 3 is not a fraction in that there is nothing but zeros after the
decimal). Carbon can be thought of in terms of the regular hexagon because it describes the
closest packing of equal radius spheres (a so-called “six-around-one”):!
But though carbon may be six protons, it has six neutrons giving it
a molar mass of 12.01 approximately twice it atomic number of
six. But twelve is our dodecagon which inscribes silicon if it is to
line up with carbon as the regular octagon. And here it is
interesting to note that carbon is made in stars by combining
beryllium-8 with helium, the eight of our regular octagon.!
Silicon is below carbon in group 14 and below that is germanium. Germanium is the other
primary semiconductor element used in transistor technology. In fact it was the first one used
and is called the first generation semiconductor where silicon is called a second generation
semiconductor and is what we mostly use today. Germanium is element 32 and has a molar
mass of 72.64. We have 72.64-32=40.64 giving it 40 neutrons. In considering the weighted
average of germanium most atoms have 42 neutrons (germanium-74). Ge consists of 5
isotopes:!
a =
s/2
tan(θ /2)
=
0.5
ta n(15
∘
)
= 1.866
Si
C
=
0.118
0.077
= 1.532
a
Si
a
C
=
1.866
1.2071
= 1.54585
1.532
1.54585
100 = 99
of 18 32
!
The electronic configuration of germanium is and has a radius of
0.137nm and a Vander Walls radius of 0.211nm. We can predict the atomic numbers of carbon,
silicon, germanium, tin…as we move down group 14 with mathematical patterns combined
with logical statements:!
If n<2!
{!
(n=1,2,3,….)!
!
If n>1; i<3!
{!
(n=2, 3. 4….). (i=1, 2, 3,…)!
!
!
For!
{!
(n=2,3,4,…)!
!
(Ar)(3d )
10
(4s)
2
(4p)
2
Z = 2
n
+ 2
n+1
Z = 2
1
+ 2
2
= 2 + 4 = 6 = carbon
Z = i ⋅ 2
n
+ 2
n+1
Z = 1 ⋅ 2
2
+ 2
3
= 4 + 8 = 12 = silicon
Z = 2 ⋅ 2
3
+ 2
4
= 2 ⋅ 8 + 16 = 16 + 16 = 32 = ger manium
Z = n(3
n
+ 2
n+2
)
Z = 2(3
2
+ 2
4
) = (9 + 16) = 50 = tin
of 19 32
If carbon is octagon (8), silicon dodecagon (12) then germanium is 16-gon. We have…!
!
!
a=2.513669746. !
!
!
a=2.513669646, Ge=0.137nm!
If a1 is octagon a1=1.2071 and a2 is 2.51367 (16-gon) then…!
!
Which is approximately chlorine (Cl=35.45g/mol). If a1 is octagon a1=1.2071 and a2 is
dodecagon a2=1.866 then…!
!
Which is approximately titanium Ti=47.88 then if we average these we have!
!
This is approximately the atomic number of calcium (Ca=40.08 g/mol). Calcium is the primary
component of the mineral component of bone, hydroxyapatite (HA). It does indeed have a
connection to germanium and silicon the primary components of AI circuitry. If we look at the
data for HA, Si, and Ge in terms of density we have!
Density of silicon is Si=2.33 grams per cubic centimeter.
Density of germanium is Ge=5.323 grams per cubic centimeter.
Density of hydroxyapatite is HA=3.00 grams per cubic centimeter.
This is
a =
(s/2)
ta n(θ /2)
=
0.5
ta n(11.25
∘
)
360
16
= 22.5
22.5
2
= 11.25
a
2
= 6.318535592
a
2
+ (s /2)
2
= 2.62915448
D = 5.125830895
P/D = 3.121445152 ≈ π
Ge
x
=
a
2
a
1
x = 72.64
1.2071
2.51367
= 34.88275868g /m ol
x = 72.64
1.2071
1.866
= 46.99021651 ≈ 47g /mol
34.88275868 + 46.99021651
2
= 40.9364876 ≈ = 41g/m ol
of 20 32
where
Where HA is the mineral component of bone, Si is an AI semiconductor material and Ge is an AI
semiconductor material. This means
The harmonic mean between Si and Ge is HA,…
This is the sextic,…
Which has a solution
Where x=Si, and y=Ge. It works for density and molar mass. It can be solved with the online Wolfram
Alpha computational engine. But,…
3
4
Si +
1
4
G e ≈ H A
H A = Ca
5
(PO
4
)
3
OH
Si
H A
Si +
[
1 −
Si
H A
]
G e = H A
2 SiG e
Si + Ge
≈ H A
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
G e
=
1
2 + 1
Si ≈ G e − H A
H A ≈
2 SiG e
Si + G e
Si ≈ G e −
2 SiG e
Si + G e
(Si + G e)G e
Si + G e
−
(Si + G e)Si
Si + G e
−
2 SiG e
Si + G e
= 0
G e
2
− 2SiG e − Si
2
Si + G e
= 0
x
2
− 2x y − y
2
= 0
x
2
− 2x y = y
2
x
2
− 2x y + y
2
= 2y
2
(x − y)
2
= 2y
2
x − y =
±
2y
of 21 32
And we find this is even more accurate by molar mass:
93% accuracy
We actually find
Si/Ge=
98.78% accuracy
The mineral component of bone hydroxyapatite (HA) is
The organic component of bone is collagen which is
We have
x = y + 2y
x = y(1 + 2)
x
y
= 1 + 2
y
x
=
1
2 + 1
Si
G e
≈
1
2 + 1
Si
G e
=
28.085
72.64
= 0.386632709
1
2 + 1
= 0.4142
0.386632709 ≈ 1 − ϕ = 0.381966011
Ca
5
(PO
4
)
3
OH = 502.32
g
m ol
C
57
H
91
N
19
O
16
= 1298.67
g
m ol
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
= 0.386795722
ϕ = 0.618033989
1 − ϕ = 0.381966011
of 22 32
98.75% accuracy
Si/Ge~
An accuracy of 99.95863%
They are almost exactly the same!
10. Silicon and Carbon
We guess that artificial intelligence (AI) has the golden ratio, or its conjugate in its means
geometric, harmonic, and arithmetic by molar mass by taking these means between doping
agents phosphorus (P) and boron (B) divided by semiconductor material silicon (Si) :
Which can be written
We see that the biological elements, H, N, C, O compared to the AI elements P, B, Si is the
golden ratio conjugate (phi) as well:
So we can now establish the connection between artificial intelligence and biological life:
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
PB
Si
=
(30.97)(10.81)
28.09
= 0.65
2 PB
P + B
1
Si
=
2(30.97)(10.81)
30.97 + 10.81
1
28.09
= 0.57
0.65 + 0.57
2
= 0.61 ≈ ϕ
PB(P + B) + 2PB
2(P + B)Si
≈ ϕ
C + N + O + H
P + B + Si
≈ ϕ
(P + B + Si )
PB(P + B) + 2PB
2(P + B)Si
≈ (C + N + O + H )
of 23 32
Which can be written:
Where HNCO is isocyanic acid, the most basic organic compound. We write in the arithmetic
mean:
Which is nice because we can write in the second first generation semiconductor as well
(germanium) and the doping agents gallium (Ga) and arsenic (As):
Where
Where ZnSe is zinc selenide, an intrinsic semiconductor used in AI, meaning it doesn’t require
doping agents. We now have:
11. Germanium And Carbon
We could begin with semiconductor germanium (Ge) and doping agents gallium (Ga) and
Phosphorus (P) and we get a similar equation:
,
In grams per mole. Then we compare these molar masses to the molar masses of the
semiconductor material Ge:
PB
[
P
Si
+
B
Si
+ 1
]
+
2 PB
P + B
[
P
Si
+
B
Si
+ 1
]
≈ 2HCNO
[
PB +
2 PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ 3HNCO
[
PB +
2 PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ HNCO
[
Ga
Ge
+
As
Ge
+ 1
]
Zn
Se
≈
[
P
Si
+
B
Si
+ 1
]
[
Ga
Ge
+
As
Ge
+ 1
]
PB
(
Zn
Se
)
+
2 PB
P + B
(
Zn
Se
)
+
P + B
2
(
Zn
Se
)
≈ HNCO
2Ga P
Ga + P
= 42.866
Ga P = 46.46749
2Ga P
Ga + P
1
Ge
=
42.866
72.61
= 0.59
of 24 32
Then, take the arithmetic mean between these:
We then notice this is about the golden ratio conjugate, , which is the inverse of the golden
ratio, . . Thus, we have
1.
2.
This is considering the elements of artificial intelligence (AI) Ga, P, Ge, Si. Since we want to find
the connection of artificial intelligence to biological life, we compare these to the biological
elements most abundant by mass carbon (C), hydrogen (H), nitrogen (N), oxygen (O),
phosphorus (P), sulfur (S). We write these CHNOPS (C+H+N+O+P+S) and find:
A similar thing can be done with germanium, Ge, and gallium, Ga, and arsenic, As, this time
using CHNOPS the most abundant biological elements by mass:
Ga P
1
Ge
=
46.46749
72.61
= 0.64
0.59 + 0.64
2
= 0.615
ϕ
Φ
ϕ ≈
1
Φ
Ga P(G a + P) + 2G a P
2(Ga + P)Ge
≈ ϕ
Ga P(G a + P) + 2G a P
2(Ga + P)Si
≈ Φ
CHNOPS
Ga + As + G e
≈
1
2
[
Ga As +
2Ga As
Ga + As
+
Ga + As
2
][
Ga
Ge
+
As
Ge
+ 1
]
≈ CHNOPS
[
Ga
Si
+
As
Si
+ 1
]
Ga As
(
O
S
)
+
2Ga As
Ga + As
(
O
S
)
+
Ga + As
2
(
O
S
)
≈ CHNOPS
O
S
≈
[
Ga
Ge
+
As
Ge
+ 1
]
[
Ga
Si
+
As
Si
+ 1
]
Ga As(Ga + As) + 2G a As
2(Ga + As)Ge
≈ 1
of 25 32
We can also make a construct for silicon doped with gallium and phosphorus:
And for germanium doped with gallium and phosphorus:
The Dynamic Construct
Above we see the artificial intelligence (AI) elements pulled out of the periodic table of the elements. As
you see we can make a 3 by 3 matrix of them and an AI periodic table. Silicon and germanium are in
group 14 meaning they have 4 valence electrons and want 4 for more to attain noble gas electron
C + H + N + O + P + S
Ga + As + G e
≈
1
2
(C + N + O + H ) ≈
2(Ga + P)Si
Ga P(G a + P) + 2G a P
(P + B + Si )
HNCO ≈
2(Ga + P)Si
(Ga + P)
[
Ga P +
2GaP
Ga + P
]
(P + B + Si )
HNCO ≈
2(P + B + Si )Si
Ga P +
2GaP
Ga + P
Ga P(G a + P) + 2G a P
2(Ga + P)Ge
≈ ϕ
[
Ga P +
2Ga P
Ga + P
+
Ga + P
2
][
P
Ge
+
B
Ge
+
Si
Ge
]
≈ HNCO
[
Ga
Ge
+
As
Ge
+ 1
]
Ga P
(
B
S
)
+
2Ga P
Ga + P
(
B
S
)
+
Ga + P
2
(
B
S
)
≈ HNCO
of 26 32
configuration. If we dope Si with B from group 13 it gets three of the four electrons and thus has a
deficiency becoming positive type silicon and thus conducts. If we dope the Si with P from group 15 it
has an extra electron and thus conducts as well. If we join the two types of silicon we have a
semiconductor for making diodes and transistors from which we can make logic circuits for AI.
As you can see doping agents As and Ga are on either side of Ge, and doping agent P is to the right of Si
but doping agent B is not directly to the left, aluminum Al is. This becomes important. I call (As-Ga) the
differential across Ge, and (P-Al) the differential across Si and call Al a dummy in the differential because
boron B is actually used to make positive type silicon.
That the AI elements make a three by three matrix they can be organized with the letter E with subscripts
that tell what element it is and it properties, I have done this:
Thus E24 is in the second row and has 4 valence electrons making it silicon (Si), E14 is in the first row
and has 4 valence electrons making it carbon (C). I believe that the AI elements can be organized in a 3 by
3 matrix makes them pivotal to structure in the Universe because we live in three dimensional space so
the mechanics of the realm we experience are described by such a matrix, for example the cross product.
Hence this paper where I show AI and biological life are mathematical constructs and described in terms
of one another.
We see, if we include the two biological elements in the matrix (E14) and and (E15) which are carbon and
nitrogen respectively, there is every reason to proceed with this paper if the idea is to show not only are
the AI elements and biological elements mathematical constructs, they are described in terms of one
another. We see this because the first row is ( B, C, N) and these happen to be the only elements that are
not core AI elements in the matrix, except boron (B) which is out of place, and aluminum (Al) as we will
see if a dummy representative, makes for a mathematical construct, the harmonic mean. Which means we
have proved our case because the first row if we take the cross product between the second and third rows
are, its respective unit vectors for the components, meaning they describe them!
12. The Computation
E
13
E
14
E
15
E
23
E
24
E
25
E
33
E
34
E
35
A = (Al, Si, P )
B = (G a, G e, A s)
A ×
B =
B
C
N
Al Si P
G a Ge As
= (Si ⋅ A s − P ⋅ G e)
B + (P ⋅ G a − Al ⋅ A s)
C + (Al ⋅ G e − Si ⋅ G a)
N
A ×
B = − 145
B + 138
C + 1.3924
N
A = 26.98
2
+ 28.09
2
+ 30.97
2
= 50g /m ol
of 27 32
And silicon (Si) is at the center of our AI periodic table of the elements. We see the biological elements C
and N being the unit vectors are multiplied by the AI elements, meaning they describe them! But we have
to ask; Why does the first row have boron in it which is not a core biological element, but is a core AI
element? The answer is that boron is the one AI element that is out of place, that is, aluminum is in its
place. But we see this has a dynamic function.
13. The Dynamic Function
The primary elements of artificial intelligence (AI) used to make diodes and transistors, silicon (Si) and
germanium (Ge) doped with boron (B) and phosphorus (P) or gallium (Ga) and arsenic (As) have an
asymmetry due to boron. Silicon and germanium are in group 14 like carbon (C) and as such have 4
valence electrons. Thus to have positive type silicon and germanium, they need doping agents from group
13 (three valence electrons) like boron and gallium, and to have negative type silicon and germanium they
need doping agents from group 15 like phosphorus and arsenic. But where gallium and arsenic are in the
same period as germanium, boron is in a different period than silicon (period 2) while phosphorus is not
(period 3). Thus aluminum (Al) is in boron’s place. This results in an interesting equation.
The differential across germanium crossed with silicon plus the differential across silicon crossed with
germanium normalized by the product between silicon and germanium is equal to the boron divided by
the average between the germanium and the silicon. The equation has nearly 100% accuracy (note: using
an older value for Ge here, is now 72.64 but that makes the equation have a higher accuracy):
=
0.213714502
2(10.81)/(72.64+28.085)=0.214643832
% accuracy
B = 69.72
2
+ 72.64
2
+ 74.92
2
= 126g /m ol
A ⋅
B = A Bcosθ
cosθ =
6241
6300
= 0.99
θ = 8
∘
A ×
B = A Bsin θ = (50)(126)sin8
∘
= 877.79
877.79 = 29.6g /m ol ≈ Si = 28.09g /m ol
Si(A s − G a) + G e(P − Al )
SiG e
=
2B
Ge + Si
(28.085)(74.9216 − 69.723) + 72.64(30.97376200 − 26.981539)
(28.085)(72.64)
0.213714502
0.214643832
= 99.567
of 28 32
Thus, due to an asymmetry in the periodic table of the elements due to boron we have the
harmonic mean between the semiconductor elements (by molar mass):
This is Stokes Theorem if we approximate the harmonic mean with the arithmetic mean:
We can make this into two integrals:
If in the equation (The accurate harmonic mean form):
We make the approximation
Then the Stokes form of the equation becomes
Thus we see for this approximation there are two integrals as well:
Si
B
(As − G a) +
Ge
B
(P − Al ) =
2SiGe
Si + G e
∫
S
( ∇ × u ) ⋅ d S =
∮
C
u ⋅ d r
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d xd y ≈
1
Ge − Si
∫
Ge
Si
x d x
∫
1
0
∫
1
0
Si
B
(As − G a)d yd z ≈
1
3
1
(Ge − Si )
∫
Ge
Si
x d x
∫
1
0
∫
1
0
Ge
B
(P − Al )d xdz ≈
2
3
1
(Ge − Si )
∫
Ge
Si
yd y
Si
B
(As − G a) +
Ge
B
(P − Al ) =
Ge − Si
Ge
Si
dx
x
2SiGe
Si + Ge
≈ Ge − Si
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d yd z =
∫
Ge
Si
d x
of 29 32
14. Conclusion
We have
Where
Gold and Iron are the quintessential elements in that gold is the finest conductor at extreme temperatures,
has had the highest value as a metal for ceremonial crafts and representation of money, since ancient
times and iron has defined the Iron Age where metallurgy and tool making reached its height. Silicon,
germanium, copper, and gold are at the crux electrical and transistor technologies as well as phosphorus,
boron, gallium and arsenic. Their mathematical representations are here connected to the stars and open
up into three dimensional geometry in the form of Stokes theorem as five-fold symmetry. The biological
elements are mathematical constructs as well and describe them. Like the interval of a fifth, musically, it
is .
∫
1
0
∫
1
0
Si
B
(As − G a)d yd z =
1
3
∫
Ge
Si
dz
∫
1
0
∫
1
0
Ge
B
(P − Al )d yd z =
2
3
∫
Ge
Si
dz
∫
1
0
∫
1
0
(
Si
B
(As − G a) +
Ge
B
(P − Al )
)
d yd z ≈
5
∫
Ge
Si
∫
Ge
Si
∇ ×
v ⋅ d
a
v = (−CP ⋅ y, SiG e ⋅ z, G a As ⋅ y)
d
a =
(
zd yd z
i + yd ydz
k
)
5
5
∏
i=1
x
i
≈ Fe
5
Si ⋅ G e ⋅ C ⋅ Ag ⋅ Au ≈ Fe
L = M
Au
Fe
Φ
of 30 32
Appendix 1
Ge=72.64!
As=74.9216!
Si=28.085!
Ga=69.723!
C=12.011!
P=30.97376200!
=!
=!
!
!
!
!
!
=154,082,837.980+16,452,521.6822=!
!
=!
(28.085)(72.64)(12.085)(107.8682)(196.9657)=!
!
Where we have substituted carbon C=12.01 for copper Cu. We use Cu, Ag, Au because they
are the middle column of our electronics matrix, they are the finest conductors used for
(Ge ⋅ As − Si ⋅ Ga)
i + (C ⋅ P)
k
[
(72.64)(74.9216) − (28.085)(69.723)
]
i +
[
(12.011)(30.97376200)
]
k
3,484.134569
(
g
mol
)
2
i + 372.025855
(
g
mol
)
2
k
∫
Ge
Si
∫
Ge
Si
(
3,484.134569
(
g
mol
)
2
i + 372.025855
(
g
mol
)
2
k
)
⋅
(
zdydz
i + yd ydz
k
)
∫
Ge
Si
∫
Ge
Si
(
3,484.134569
(
g
mol
)
2
⋅ zdzdy + 372.025855
(
g
mol
)
2
⋅ yd zd y
)
∫
Ge
Si
3,484.134569
(
(72.64 − 28.085)
2
2
)
dy +
∫
Ge
Si
372.025855y ⋅ (72.64 − 28.085)d y
3458261.42924
(
g
mol
)
4
(72.64 − 28.085) + 16575.6119695
(
g
mol
)
3
(
(72.64 − 28.085)
2
2
)
170,535,359.662
(
g
mol
)
5
5
∏
i=1
x
i
= Si ⋅ Ge ⋅ C ⋅ Ag ⋅ Au
523,818,646.5
g
5
mol
5
of 31 32
electrical wire. We use C, Si, Ge because they are the middle column of our AI Biomatrix. Si
and Ge are the primary semiconductor elements used in transistor technology (Artificial
Intelligence) and C is the core element of biological life. We have!
!
!
Perimeter/Diameter of regular hexagon = 3.00!
!
The same value as our 3.0716 if taken at two places after the decimal.!
523,818,646.5
170,535,359.662
= 3.0716
π = 3.141...
3.141 + 3.00
2
= 3.0705
of 32 32
The Author!